Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?pptrf
Computes the Cholesky factorization of a symmetric (Hermitian) positive-definite matrix using packed storage.
Syntax
call spptrf ( uplo , n , ap , info )
call dpptrf ( uplo , n , ap , info )
call cpptrf ( uplo , n , ap , info )
call zpptrf ( uplo , n , ap , info )
call pptrf ( ap [ , uplo ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
spptrf dpptrf cpptrf zpptrf pptrf
The routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian positive-definite packed matrix A :
\(A = U^{T}*U\) for real data, \(A = U^{H}*U\) for complex data |
if uplo='U' |
\(A = L*L^{T}\) for real data, \(A = L*L^{H}\) for complex data |
if uplo='L' |
where \(L\) is a lower triangular matrix and \(U\) is upper triangular.
This routine supports the Progress Routine feature. See Progress Function for details.
Input Parameters
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ .
Indicates whether the upper or lower triangular part of \(A\) is packed in the array ap , and how \(A\) is factored:
If uplo = ‘U’ , the array ap stores the upper triangular part of the matrix \(A\) , and \(A\) is factored as \(U^H*U\) .
If uplo = ‘L’ , the array ap stores the lower triangular part of the matrix A , and \(A\) is factored as \(L*L^H\) .
n
INTEGER . The order of matrix A ; n ≥ 0.
ap
REAL for spptrf
DOUBLE PRECISION for dpptrf
COMPLEX for cpptrf
DOUBLE COMPLEX for zpptrf .
Array, size at least max(1, n(n+1)/2). The array ap contains either the upper or the lower triangular part of the matrix A (as specified by uplo ) in packed storage (see Matrix Storage Schemes ).
Output Parameters
ap
Overwritten by the Cholesky factor U or L , as specified by uplo .
info
INTEGER .
If info=0 , the execution is successful.
If info = -i , the i- th parameter had an illegal value.
If info = i , the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the factorization could not be completed. This may indicate an error in forming the matrix A .
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions .
Specific details for the routine pptrf interface are as follows:
ap
Holds the array A of size (n*(n+1)/2) .
uplo
Must be ‘U’ or ‘L’ . The default value is ‘U’ .
Application Notes
If uplo='U', the computed factor \(U\) is the exact factor of a perturbed matrix \(A + E\) , where
\[|E| \leq c(n) \epsilon |U^H| |U|, \quad |e_{ij}| \leq c(n) \epsilon \sqrt{a_{ii}a_{jj}}\]
c(n) is a modest linear function of n , and \(\varepsilon\) is the machine precision.
A similar estimate holds for uplo = 'L' .
The total number of floating-point operations is approximately \((1/3)n^{3}\) for real flavors and \((4/3)n^{3}\) for complex flavors.
After calling this routine, you can call the following routines:
?pptrs (Solves a system of linear equations with a packed Cholesky-factored symmetric (Hermitian) positive-definite coefficient matrix.)
to solve \(AX = B\)
?ppcon (Estimates the reciprocal of the condition number of a packed symmetric (Hermitian) positive-definite matrix.)
to estimate the condition number of \(A\)
?pptri (Computes the inverse of a packed symmetric (Hermitian) positive-definite matrix using Cholesky factorization.)
to compute the inverse of \(A\) .